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Gza
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Could someone give me an intuitive example of extrinsic and intrinsic curvature. That would be much appreciated, thanks in advance.
Gza said:Yes, that was very helpful. I had to dig up a softer book on a treatment of tensors, but it still served its purpose, thanks again.
The cylinder is an excellant example of zero curvature. It is also an excellant example of a manifold for which there are infinitely many geodesics between any two points on the surface.
Gza said:Maybe I'm referring to the wrong concept, but I thought a circle had a curvature inverse of its radius, so wouldn't the curved part of the cylinder have curvature?
Extrinsic curvature refers to the curvature of a surface in three-dimensional space. It is a measure of how a surface is curved in relation to its embedding in a higher-dimensional space.
Intrinsic curvature refers to the curvature of a surface as it exists within its own two-dimensional space. It is a measure of how the surface is curved without considering its embedding in a higher-dimensional space.
The main difference between extrinsic and intrinsic curvature is that extrinsic curvature takes into account the surface's embedding in a higher-dimensional space, while intrinsic curvature only considers the curvature of the surface itself.
Extrinsic curvature is typically measured using differential geometry, specifically using the Gauss-Codazzi equations. This involves calculating the first and second fundamental forms of the surface, which describe the surface's local geometry.
Extrinsic and intrinsic curvature are important concepts in mathematics, physics, and engineering. They have applications in fields such as differential geometry, general relativity, and computer graphics. Understanding these concepts allows scientists to better understand the shape and behavior of surfaces in different dimensions.